Properties

Label 2-1344-168.149-c0-0-2
Degree $2$
Conductor $1344$
Sign $0.980 + 0.197i$
Analytic cond. $0.670743$
Root an. cond. $0.818989$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)3-s + (0.866 + 0.5i)7-s + (0.499 − 0.866i)9-s + 1.73i·13-s + (−0.866 − 0.5i)19-s + 0.999·21-s + (0.5 + 0.866i)25-s − 0.999i·27-s + (−0.866 − 1.5i)31-s + (−1.5 − 0.866i)37-s + (0.866 + 1.49i)39-s i·43-s + (0.499 + 0.866i)49-s − 0.999·57-s + (0.866 − 0.499i)63-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)3-s + (0.866 + 0.5i)7-s + (0.499 − 0.866i)9-s + 1.73i·13-s + (−0.866 − 0.5i)19-s + 0.999·21-s + (0.5 + 0.866i)25-s − 0.999i·27-s + (−0.866 − 1.5i)31-s + (−1.5 − 0.866i)37-s + (0.866 + 1.49i)39-s i·43-s + (0.499 + 0.866i)49-s − 0.999·57-s + (0.866 − 0.499i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.980 + 0.197i$
Analytic conductor: \(0.670743\)
Root analytic conductor: \(0.818989\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :0),\ 0.980 + 0.197i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.539140871\)
\(L(\frac12)\) \(\approx\) \(1.539140871\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
good5 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - 1.73iT - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + iT - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + 2T + T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.305375519762864375978932383853, −9.040180286002621190107320790153, −8.281563512959628639286162862746, −7.31776831144826739565169662696, −6.77261580839735768819170297878, −5.66656599874951787414939501177, −4.52946621521874778386516169117, −3.73619799099786836484352427627, −2.32637257126767035231021023640, −1.71936963225688645783096004183, 1.56163922652239484991610100398, 2.84874015659539083664884360037, 3.71094013236821201846506839146, 4.73069243349022429596076097372, 5.38537914834218028578613808840, 6.71263271333190800662654143049, 7.70797342041794686363990231207, 8.257949204544708563433343436772, 8.809403759642098163147127063327, 10.00698876290157562610644250422

Graph of the $Z$-function along the critical line